$\dfrac{ 8e + 6f }{ -6 } = \dfrac{ -6e - 3g }{ -4 }$ Solve for $e$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ 8e + 6f }{ -{6} } = \dfrac{ -6e - 3g }{ -4 }$ $-{6} \cdot \dfrac{ 8e + 6f }{ -{6} } = -{6} \cdot \dfrac{ -6e - 3g }{ -4 }$ $8e + 6f = -{6} \cdot \dfrac { -6e - 3g }{ -4 }$ Multiply both sides by the right denominator. $8e + 6f = -6 \cdot \dfrac{ -6e - 3g }{ -{4} }$ $-{4} \cdot \left( 8e + 6f \right) = -{4} \cdot -6 \cdot \dfrac{ -6e - 3g }{ -{4} }$ $-{4} \cdot \left( 8e + 6f \right) = -6 \cdot \left( -6e - 3g \right)$ Distribute both sides $-{4} \cdot \left( 8e + 6f \right) = -{6} \cdot \left( -6e - 3g \right)$ $-{32}e - {24}f = {36}e + {18}g$ Combine $e$ terms on the left. $-{32e} - 24f = {36e} + 18g$ $-{68e} - 24f = 18g$ Move the $f$ term to the right. $-68e - {24f} = 18g$ $-68e = 18g + {24f}$ Isolate $e$ by dividing both sides by its coefficient. $-{68}e = 18g + 24f$ $e = \dfrac{ 18g + 24f }{ -{68} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $e = \dfrac{ -{9}g - {12}f }{ {34} }$